Let $(M,g)$ be a fixed closed Riemannian manifold, normalized to have volume 1. We'll write $d_M(x,y)$ for the (geodesic) distance between two points $x,y\in M$. I'm interested in the following class of functions $\varphi: M\to \mathbb{R}$.

In this case, we say that $\varphi$ is the generalized Legendre transform of $\psi$, which we write $\varphi = \psi^c$

In particular, one can show that if a $d^2/2$-convex function is bounded, Lipschitz (and thus differentiable $vol_M$ a.e.) with gradient bounded by the diameter of $M$. Also, $(\varphi^c)^c=\varphi$ if and only if $\varphi$ is $d^2/2$ convex.

I'm interested in these because of their relation to the theory of optimal transport. I'll briefly describe it here, but in theory one should not need to know anything about optimal transport to answer my questions (although it may help as it seems intimately related). I'm intersted in the 2-Wasserstein metric on $\mathcal{P}(M)$, the space of probability measures on $M$. Lying a bit, I'll say that this is defined to be, for $\mu,\nu\in \mathcal{P}(M)$ probability measures
$$ d^W(\mu,\nu)^2 : = \inf_{F:M\to M, F_*\mu = \nu} \int_M d(x,F(x))^2 dvol_M$$
Here, $F_*\mu = \nu$ means that for all Borel sets $A$, $\mu(F^{-1}(A)) = \nu(A)$. (In fact this is only really true when $\mu$ and $\nu$ are absolutely continuous wrt the volume measure, and it needs a bit of generalization to be really true. Anyways, what is relevant to the beginning of the question, is that for any measure $\nu \in \mathcal{P}(M)$, there is a unique (up to constants) $d^2/2$-convex function $\varphi$ such that $\nu = exp(\nabla \varphi)_* vol_M$, and it turns out this is the unique minimizer in the above definition of distance with $\mu = vol_M$. That is
$$
d^W(vol_M,\nu) = \int_M |\nabla \varphi|_g^2 d vol_M
$$

Furthermore, the geodesic from $vol_M$ to $\nu$ in $\mathcal{P}(M)$ (in the metric space sense) is given by $t\mapsto \exp(t\nabla \varphi)_* vol_M$.

Now in this paper, Sturm (not sure if he was the first to do this) shows that this gives a continuous involution on $\mathcal{P}(M)$, by taking the Legendre transform of the $d^2/2$-convex $\varphi$ (uniquely) associated to $\nu$ and setting $\nu^c := \exp(\nabla \varphi^c)_* vol_M$.

I'm interested in various geometric properties related to this involution, which can be rephrased in elementary terms as follows (I've included my geometric interpretations in quotations):

Is it true that for $t\in[0,1]$, $(t\varphi)^c = t (\varphi^c)$?
"geodesics from $vol_M$ are mapped to other geodesics from $vol_M$"
(I think I can prove this using some weird scaling arguments, but I'd like an elementary proof just from the definition, which should probably exist if it is true)

What is the relation (if any) between
$$
\int_M |\nabla \varphi|_g^2 dvol_M
$$
and
$$
\int_M |\nabla \varphi^c|_g^2 dvol_M
$$
"how close are $d^W(vol_M,\nu)$ and $d^W(vol_M,\nu^c)$?"

For any $d^2/2$-convex function $\varphi$, let $M$ be the supremum of $m$ such that $m\varphi$ is $d^2/2$-convex. One can show that $M\varphi$ is then $d^2/2$-convex (the set of $d^2/2$-convex functions is closed in $H^1$, see the Sturm paper). What does $\exp(M\nabla \varphi)_* vol_M$ look like? Is it in general totally singular, etc? "what do the endpoints of geodesics starting from $vol_M$ look like?"

How should I think of this map geometrically, i.e. on the level of measures?

I'm most interested in (2), followed by (3), but have included (4) just in case someone has any insight - as far as I can tell this is not very well understood and probably does not have a good answer right now.

1 Answer
1

Concerning (2), if I don't mix up notations and when $\nu$ is absolutely continuous, $x\mapsto x+\nabla\varphi^c$ is the Brenier map from $\nu$ to $vol_M$. As a consequence, it holds $d^W(vol_M,\nu)=\int_M|\nabla\varphi^c|_g^2 d\nu$, so if $\nu$ is far away from $vol_M$ and $\varphi$ is not too close to be constant, then your two integrals can be very different, and I do not see why they should have a relation.